Question

Show that the Gaussian integer \(a+b i(b \neq 0)\) is prime if and only if the \(\operatorname{norm} a^{2}+b^{2}\) is an ordinary prime.

Short answer

Short Answer: A Gaussian integer a+bi (with b≠0) is prime if and only if its norm a^2+b^2 is an ordinary prime. The proof consists of three steps: (1) define Gaussian integers, ordinary primes, and the norm function, (2) show that if a Gaussian integer is prime, then its norm is an ordinary prime, and (3) show that if the norm is an ordinary prime, then the Gaussian integer is prime.

Step-by-step solution

Define Gaussian integers, ordinary primes, and the norm function

Gaussian integers are complex numbers of the form a+bi, where a and b are integers, and i is the imaginary unit (i^2 = -1). An ordinary prime is a positive integer p greater than 1 that is only divisible by 1 and itself (e.g., 2, 3, 5, 7, etc.). The norm function for Gaussian integers is defined as N(a+bi) = a^2 + b^2.

Prove that if a Gaussian integer a+bi is prime, then its norm a^2+b^2 must be an ordinary prime

Suppose a Gaussian integer a+bi is prime. This means that it cannot be factored into Gaussian integers except for the trivial factors (units and associates) of itself. Let's compute the norm of a+bi as N(a+bi) = a^2 + b^2. Now consider the equation N(αβ) = N(α)N(β) for any Gaussian integers α, β. If a+bi has non-trivial factors, then N(a+bi) has factors N(α) and N(β) where N(α), N(β) > 1. Conversely, if this does not happen, a+bi does not have any non-trivial factors, which would imply that it is prime. Therefore, if a+bi is prime, its norm a^2+b^2 must be an ordinary prime.

Prove that if the norm a^2+b^2 is an ordinary prime, then the Gaussian integer a+bi must be prime

Suppose the norm a^2+b^2 is an ordinary prime, say p. If a+bi is not prime, it means that a+bi has non-trivial factors, say α and β, such that a+bi = α β. Using the norm equation, we have N(a+bi) = N(α)N(β). So, it means p = N(α)N(β) where N(α), N(β) > 1. But, since p is an ordinary prime, this is impossible. Because the only divisors of an ordinary prime p are 1 and p itself. Thus a+bi must be prime. In conclusion, we've shown that the Gaussian integer a+bi (where b≠0) is prime if and only if the norm a^2+b^2 is an ordinary prime.

Key concepts

Prime Numbers

Prime numbers are the building blocks of arithmetic. They are defined as natural numbers greater than 1, which only have two divisors: 1 and the number itself. This means that a number like 7 is prime, because it can only be divided by 1 and 7 without leaving a remainder.

Prime numbers are crucial in number theory due to several reasons: they are the 'atoms' of integer arithmetic and form the foundation for more complex structures.

• In the context of Gaussian integers, a primary definition of a prime still applies but is more nuanced.
• Gaussian primes are complex numbers that cannot be expressed as products of other Gaussian integers, except trivial products.
• Determining primes in other forms, like in Gaussian integers, helps advance our understanding of the integers.

Understanding what makes a number prime in different mathematical contexts, such as for Gaussian integers, enriches the theory surrounding these numbers.

Norm Function

The norm function plays a critical role in understanding the properties of Gaussian integers among other mathematical entities. For a Gaussian integer, which takes the form of a complex number, the norm function is defined as:\[N(a+bi) = a^2 + b^2\]where \(a\) and \(b\) are integers.

The essence of the norm function lies in its ability to convert complex numbers into real numbers, allowing us to compare them and assess their divisibility like regular integers. This is especially useful when we try to determine whether a Gaussian integer is prime.

• The norm of a Gaussian prime is particularly significant, as it ties into the necessity for the norm to also be an ordinary prime.
• If this calculated norm value is prime, then it indicates no smaller Gaussian divisors exist apart from the trivial ones, keeping the Gaussian integer itself as a 'prime'.
• Thus, investigating the norm is a fundamental step in identifying Gaussian primes.

For Gaussian integers, the prime characteristic is intricately linked to the norm's primality.

Complex Numbers

Complex numbers are numbers that have two components: a real part and an imaginary part. They are usually expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, with the important property\(i^2 = -1\).

These numbers extend our number system to include solutions to equations that do not have real solutions, such as \(x^2 + 1 = 0\). In Gaussian integers, which are a specific subset of complex numbers, both \(a\) and \(b\) are integers.

• Complex numbers are incredibly useful in various branches of mathematics and engineering. They allow for a more complete picture when analyzing problems that involve waves, stability in control systems, and more.
• In the context of Gaussian primes, we require complex numbers to step beyond simple integer arithmetic.
• Using complex numbers helps us explore new types of primality and relationships between numbers.

Incorporating complex numbers into exploring Gaussian integers helps bridge the real and imaginary, providing a richer understanding of mathematical interactions.